Another cool application of summation notation with matrices is to prove things about the trace of a matrix. The trace only applies to square matrices (equal number of rows and columns) and is the sum of all the entries on the diagonal-that is, the sum of all entries with the same column and row number. In summation notation, the trace of an n x n matrix as: Tr (A) = a1,1 + a2,2+...+ ann =Σaii This time we are using the index i for both the row position and the column position, so its the position of the index that denotes row and column. The formula for the product used two different letters for the indices because they were not always equal, but for trace the row and column number will always be equal, so we only need one letter. The next exercise covers some basic properties of traces: Exercise 11.7.1. (a) Prove that if A and B are square matrices of the same size, then Tr (A + B) = Tr (A) + Tr (B). (b) Prove that if A is a square matrix with real entries and k is a real number, then Tr (kA) = kTr (A).

Another cool application of summation notation with matrices is to prove things about the trace of a matrix. The trace only applies to square matrices (equal number of rows and columns) and is the sum of all the entries on the diagonal-that is, the sum of all entries with the same column and row number. In summation notation, the trace of an n x n matrix as: Tr (A) = a1,1 + a2,2+...+ ann =Σaii This time we are using the index i for both the row position and the column position, so its the position of the index that denotes row and column. The formula for the product used two different letters for the indices because they were not always equal, but for trace the row and column number will always be equal, so we only need one letter. The next exercise covers some basic properties of traces: Exercise 11.7.1. (a) Prove that if A and B are square matrices of the same size, then Tr (A + B) = Tr (A) + Tr (B). (b) Prove that if A is a square matrix with real entries and k is a real number, then Tr (kA) = kTr (A).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.2: Direct Methods For Solving Linear Systems

Problem 22EQ: Consider the matrix A=[2314]. Show that any of the three types of elementary row operations can be...

See similar textbooks

Similar questions

Find the sequence of the elementary matrices whose product is the non singular matrix below. [2410]
15. Reverse the elementary row operations used in Example 2.9 to show that we can convert
To find the product matrix AB. the number of _______________ of A must be the same as the number of _________ of B.
Consider the matrix A=[2314]. Show that any of the three types of elementary row operations can be used to create a leading 1 at the top of the first column. Which do you prefer and why?
Find two 22 matrices such that |A|+|B|=|A+B|.